Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 66654.bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66654.bt1 | 66654bw2 | \([1, -1, 1, -4205, 105049]\) | \(926859375/9604\) | \(85185011772\) | \([2]\) | \(73728\) | \(0.91437\) | |
66654.bt2 | 66654bw1 | \([1, -1, 1, -65, 4033]\) | \(-3375/784\) | \(-6953878512\) | \([2]\) | \(36864\) | \(0.56780\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66654.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 66654.bt do not have complex multiplication.Modular form 66654.2.a.bt
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.